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Personally, I don’t like this terminology, and I would prefer not to propagate it. What do others think?
I like the analogy with symmetric monoidal categories. What don’t you like about it?
That it’s a false analogy? As you wrote on the page, a “symmetric bicategory” with one object is not the same as a symmetric monoidal category.
I have made some edits:
where the term “symmetric bicategory” first appears in the text, I have added a pointer to May-Sigurdsson’s definition, also for the closed version;
have added the publication data and links for:
together with a warning that the definition is only in the published pdf-version (it’s not in the arXiv version, as far as I can see).
hyperlinked biequivalence and added the (hyperlinked) term opposite 2-category;
fixed the typesetting of – the previous $B^{\text{op}}$ does not render as desired (no superscript size), it’s $B^{\mathrm{op}}$ instead, but Instiki actually allows to just write $B^{op}$ and have the same effect.
made the remark on categorification a numbered remark;
(would want to make the definition a numbered definition, too, but I am uncertain whether what we have there is to be taken as the precise definition or as a sketch of a definition. May-Sigursson have a much stricter definition with a comment, p. 258, to the effect that they’d rather postpone thinking about its generalization);
hyperlinked the term symmetric monoidal category and added cross-link from there back to here;
changed the wording of the remark:
It is not true, in general, that “a symmetric bicategory is a categorification of a monoidal category”, and even when it happens to be the case this is not what we intend to say here: It is the notion of symmetric bicategories that is a categorification of the notion of monoidal categories.
(This is pedantic respective to common people, but should not be overly pedantic on a mathematical wiki with focus on categorical logic.)
similarly, I changed the previous “if is the identity” – which didn’t type-check – to “if the component functor of is the identiy functor on the hom-category”;
added the floating context menu for 2-category theory.
That it’s a false analogy? As you wrote on the page, a “symmetric bicategory” with one object is not the same as a symmetric monoidal category.
I appreciate that sentiment. At the same time, my feeling is that if a concept does not admit a direct horizontal categorification, there is some sense in using the same name for its closest generalisation. My understanding (possibly flawed) is that any theorem that is true for symmetric monoidal categories will be true (possibly in a less strict sense) for one-object symmetric bicategories, so, up to a suitable notion of equivalence, they’re really the same anyway, so it is not a harmful identification.
I have made some edits:
Thank you! This is a helpful example of best practice for nLab pages.
would want to make the definition a numbered definition, too, but I am uncertain whether what we have there is to be taken as the precise definition or as a sketch of a definition.
Regarding Definition 16.2.1? You’re right, I had assumed it was an equivalent definition, but given the ambiguity in the text, it would be helpful to check this.
Thanks, yes, I ended up numbering the definition after all. Maybe you could add a brief line on how this really compares to May-Sigurdsson? I’d expect your nicer+quicker+abstract definition to be the right one, but I wouldn’t know if there is any pitfall to beware of, as I haven’t really thought about it.
Regarding terminology: Since the concept has been published under that very name and apparently under that name only (?), there is little room left for debate on how to name the entry, if the entry is – as it currently is – referring to that publication. But the entry has plenty of room for a section “Terminology” into which everyone can write discussion of their private preferred terminology. If convincing, then there is a chance that readers will pick it up.
Added reference to dagger category, which is a decategorification of a symmetric bicategory.
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